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For a function f, let f*(x) = lim as h→0 be [f(x+h) - f(x-h)]/h

a) Determine f*(x) for f(x) = cosx
b) write an equation that expresses the relationship between the functions f* and f` where f` denotes the usual derivative of f

  • calculus -

    Use the trigonometric identity
    cos (x + h) = cosx cosh - sinx sinh
    This means that
    cos(x + h) - cosx
    = (cosh -1) cosx - sin x sinh
    For very small h, sin h = h and cos h = 1, so [cos(x + h) - cosx]/h -> -sin x
    -sin x is the exact derivative of cos x.

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